Exterior power operations on higher K-theory

We construct operations on higher algebraic K-groups induced by operations such as exterior power on any suitable exact category, without appeal to the plus-construction of Quillen.Dedicated to Alexander Grothendieck on his sixtieth birthdaySupported by NSF grants DMS 85-04692, 86-01980.     

Polynomial operations on complex K-theory

We adapt here the results of the author concerning polynomial operations on the 0th stable cohomotopy to the case of the 0th complex K-theory and consider polynomial operations : Kh, where h is a ring-valued contravariant functor, defined on finite CW-complexes, satisfying some properties. We construct a family of generating operations for the ring Pol(K,h) of all polynomial operations : Kh and doing so, we describe the additive structure of this ring in terms of the h(BU(n)'s. As an illustration of how polynomiality could be used to study operations in the setting of algebraic K-theory, we consider, from our point of view, the well known situation operations : KK on complex K-theory.     

Some filtrations on higherK-theory and related invariants

We lift Bloch's higher Chow construction from the level of simplicial sets to the level of simplicial spaces. We construct a simplicial space that becomes isomorphic to the Bloch/Chow complex when the functor ₀ is applied in each degree. The homotopy groups of this space are theE ₂-terms in an Atiyah-Hirzebruch spectral sequence converging to algebraicK-theory. TheseE ₂-terms map nontrivially to the expected higher Chow groups. We define and compute several intermediate invariants associated to our simplicial space.     

New relationships between steenrod operations for certain spaces

In this paper, we show that under a certain technical condition, if a space has no 2-torsion, then eitherS q ² n x≠0 or there exists somey withS q ² n y=S q ² n ⁺¹ x, if for somen≥3S q ² n ⁺¹ x≠0. The proof uses relations between Steenrod operations and operations in connective realK-Theory. This research was partially supported by NSERC.     

Adams Operations for Bivariant K-Theory and a Filtration Using Projective Lines

We establish the existence of Adams operations on the members of a filtration of K-theory which is defined using products of projective lines. We also show that this filtration induces the gamma filtration on the rational K-groups of a smooth variety over a field of characteristic zero.     

Geometric realization of Adams maps

Let G/P be a homogenous space with G a compact connected Lie group and P a connected subgroup of G of equal rank. As the rational cohomology ring of G/P is concentrated in even dimensions, for an integer k we can define the Adams map of type k to be l _k : H*(G/P, ℚ) → H*(G/P, ℚ), l _k (u) = k ⁱ u, u ∈ H ²ⁱ (G/P, ℚ). We show that if k is prime to the order of the Weyl group of G, then l _k can be induced by a self map of G/P. We also obtain results which imply the condition that k is prime to the order of the Weyl group of G is necessary.     

The MoravaK-theory of algebraicK-theory spectra

Ifn2 the MoravaK-theoryK(n) _* of an algebraicK-theory spectrumKX vanishes for any ring or schemeX. This is proved using thev _n -complexes of Hopkins and Smith, together with the following theorem. The natural mapf:Q ₀S⁰BGL⁺ factors through the space ImJ. In particularf _*: _* ^s K _* annihilates CokerJ. These results are closely related to the Lichtenbaum-Quillen conjectures.Partially supported by an NSF grant.     

A delay differential equation solver based on the parallel Adams algorithms

IntroductionWith the rapid development of science and technology, a lot of delay-dynamic systems invarious fields of natural and social science have arisentl]. The presented delay systems are oftenof both multiparameter and largescale, that is, so-called large systems. Considering scientificcomputation and real-time simulation, the classical serial algorithmsIZ--4] have not fulfilled therequirements for solying this kind of systems. So, it becomes quite emergent to construct thefast algorithm…     

ContinuousK-theory

We study arithmetical properties of homotopy groups of thel-adic completion of Quillen'sK-theory space of number field, with a view on the Dwyer-Friedlander comparison map into étaleK-theory. The relation of these groups toK-theory is a complete analogy to the relation of continuous étale cohomology to étale cohomology. We identify the torsion subgroup of the resulting term with the subgroup of divisible elements inK _2n (F). We prove that this term is sent isomorphically into étaleK-theory, giving some further evidence for the Lichtenbaum-Quillen conjectures.     

MilnorK-theory is the simplest part of algebraicK-theory

We identify the MilnorK-theory of a field with a certain higher Chow group.     

Continuously controlledA-theory

To anycontrolled space over the metric spaceB we can associate itsboundedly controlled algebraic K-theory, a functor designed to give information about the space of stable bounded concordances of manifolds homotopy equivalent toX. Generalizing a construction of D. R. Anderson, F. X. Connolly, S. Ferry, and E. K. Pedersen, we define another functor, calledcontinuously controlled A-theory, which depends only on thetopology of the control space, not itsmetric properties. In the special case whereB=R ₊, this functor is (more or less by definition) the same asproper A-theory. We prove that under certain conditions on the controlled space the natural transformation from boundedly controlledA-theory to continuously controlledA-theory is a weak homotopy equivalence, and hence defines a generalized homology theory. Continuously controlledK-theory is used in the approaches of G. Carlsson, E. K. Pedersen, and S. Ferry, S. Weinbergervto theK-theory Novikov conjecture.     

Symmetric operations in algebraic cobordism

In this article we describe certain new cohomological operations in algebraic cobordisms. These operations give the natural obstructions for the cobordism class to be represented by the embedding. Also, they permit to work with algebraic cobordisms and Chow groups in a more subtle way than the Landweber-Novikov operations (related to 2-torsion effects). We describe applications to the computation of the algebraic cobordisms of a Pfister quadrics and to the problem of rationality of cycles.     

Sobolev-Lorentz范数约束下的次临界型Adams不等式

本文在Sobolev-Lorentz空间W2L2,q(R4)的范数约束下得到了一个最佳的二阶次临界型Adams不等式.进一步,当次临界指标逼近最佳常数时,得到了Adams泛函的上、下界的估计.本文主要采用了Lam和Lu[A new approach to sharp Moser-Trudinger and Adams ...     

Relative MilnorK-theory

LetR be a commutative, semi-local ring,I ₁, ...,I _s ideals. In this paper, we define therelative Milnor K-groups of (R;I ₁, ...,I _s ),K _p ^M (R;I ₁, ...,I _s ), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic to the weightp portion of the relative QuillenK-groupK _p (R;I ₁, ...,I _s ), after inverting (p–1)!. We also define the relative group homology of GL _n (R;I ₁, ...,I _s ), and show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic toH _p (GL_p(R;I ₁, ...,I _s ))/Im(H _p (GL _p–1 (R;I ₁, ...,I _s ))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK _p ^M (F)/l ^v to , and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:

Additive L-theory

The cobordism groups of quadratic Poincaré complexes in an additive category with involution are identified with the Wall L-groups of quadratic forms and formations in , generalizing earlier work for modules over a ring with involution by avoiding kernels and cokernels.     

A pull back theorem in the Adams spectral sequence

This paper proves that, for any generator x∈ExtA^s,tq（Zp,Zp）, if （1L ∧i）＊Ф＊（x）∈ExtA^s＋1,tq＋2q（H＊L∧M, Zp） is a permanent cycle in the Adams spectral sequence （ASS）, then b0x ∈ExtA^s＋1,tq＋q（Zp, Zp） also is a permenent cycle in the ASS. As an application, the paper obtains that h0hnhm∈ExtA^3,pnq＋p^mq＋q（Zp, Zp） is a permanent cycle in the ASS and it converges to elements of order p in the stable homotopy groups of spheres πp^nq＋p^mq＋q-3S, where p ≥5 is a prime, s ≤ 4, n ≥m＋2≥4 and M is the Moore spectrum.     

Infinite sums of Adams operations and cobordism

The elements of various algebras of stable degree zero operations in p-local K-theory can be described explicitly as certain infinite sums of Adams operations [11, 9]. Here we show how to make sense of the same expressions for MU_(p) and BP, thus identifying the “Adams subalgebra” of the algebras of operations. We prove that the Adams subalgebra is the centre of the ring of degree zero operations.     

Adams谱序列上的非平凡乘积b_0k_0δ_(s+4)

主要用May谱序列证明了非平凡的乘积b_0k_0δ_(s+4)∈Ext_A~(s+8,t)(Z_p,Z_p),其中p是大于等于7的素数,0≤sp-4,q=2(p-1),t=(s+4)p~3q+(s+3)p~2q+(s+5)pq+(s+2)q+s.     

Omnichannel operations with ship-from-store

While embracing omnichannel operations, firms are rolling out ship-from-store (SFS). This paper considers a newsvendor-like retailer operating both online and offline channels, and analyzes how SFS affects the seller’s profits. In our model, consumers strategically choose a channel to optimize their utility, which is investigated by applying the theory of hyperbolic discounting. The results show that, interestingly, the firm can be worse off implementing SFS, when the customer-base expansion thanks to SFS is not significant.     

Mackey structures on equivariant algebraic K-theory

We study the Mackey structure of the G-spectrum K _G C associated to a monoidal G-category C. It is proved that the coefficient system of K _G C coincides, as a (graded) Mackey functor, with the system of equivariant K-groups in the sense of Fröhlich and Wall. It is also shown that for any exact category U, there exists a G-spectrum Q _G U representing the equivariant K-theory of U in the sense of Dress and Kuku, and that Q _G U is naturally G-homotopy equivalent to K _G IsoU if every short exact sequence in U splits.